Suppose we have a right triangle with integer sides (in some unit of measure). Prove
(a) One of the legs has a length divisible by 3.
(b) One of the three sides has length divisible by 5.
We want to show either $\displaystyle m^2-n^2 $ or $\displaystyle 2mn $ is divisible by $\displaystyle 3 $.
WLOG if $\displaystyle m\equiv0\bmod{3} $ then we're done since then $\displaystyle 2mn\equiv0\bmod{3} $.
So assume $\displaystyle m $ and $\displaystyle n $ are both not divisible by $\displaystyle 3 $.
But then by flt we get that $\displaystyle m^2\equiv1\bmod{3} $ and $\displaystyle n^2\equiv1\bmod{3} $, so $\displaystyle m^2-n^2\equiv0\bmod{3} $.